As usual, write \(\mathbf {s}_{k-1} = \mathbf {x}_k-\mathbf {x}_{k-1}\)  and \(\mathbf {y}_{k-1} = \mathbf {g}_k-\mathbf {g}_{k-1}\) .
Two popular stepsizes are the Barzilai–Borwein (BB) steplengths [1]}
\(\beta _{k}^{\rm {BB1}} = \frac{\mathbf {s}_{k-1}^T \mathbf {s}_{k-1}}{\mathbf {y}_{k-1}^T \mathbf {s}_{k-1}}, \qquad \beta _{k}^{\rm {BB2}} = \frac{\mathbf {y}_{k-1}^T \mathbf {s}_{k-1}}{\mathbf {y}_{k-1}^T \mathbf {y}_{k-1}}.\) 
